Circle theorems and their applications | Year 10

01 / LEARN

Understand the idea.

Four vertices on one circle

A cyclic quadrilateral has all four vertices on one circle. Its opposite interior angles are supplementary: each uses half an intercepted arc, and the two opposite arcs together cover the circle.

Match opposite vertices before subtracting from $180^{\circ}$ ; neighbouring angles do not generally have this sum. The converse can establish cyclicity for a simple quadrilateral when a pair of opposite interior angles is supplementary.

∠ A + ∠ C = 180^{\circ}

Tangent and chord

The tangent–chord theorem matches an angle between a tangent and a chord to the circumference angle in the alternate segment. Locate the tangent’s contact point and the chord’s other endpoint before choosing the matching angle.

A tangent line forms two supplementary angles with a chord. Use the stated angle and alternate segment to identify the correct one. Only the radius to the contact point is guaranteed perpendicular to the tangent.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Cyclic quadrilateral

ABCD is cyclic in that order. ∠A=80° and ∠B=110°. Find ∠C and ∠D.

Worked solution

1
Pair the opposite vertices.
$∠ A + ∠ C = 180^{\circ}, ∠ B + ∠ D = 180^{\circ}$
The cyclic order ABCD makes A opposite C and B opposite D.
2
Calculate angle C.
$∠ C = 180^{\circ} - 80^{\circ} = 100^{\circ}$
Opposite interior angles of a cyclic quadrilateral are supplementary.
3
Calculate angle D.
$∠ D = 180^{\circ} - 110^{\circ} = 70^{\circ}$
Apply the same theorem to the other opposite pair.

Answer

100° and 70°

EXAMPLE 02

Tangent and chord

The acute angle between a tangent at A and chord AB is 52°. C is on the alternate segment. Find ∠ACB.

Worked solution

1
Match chord AB to the alternate-segment angle.
Angle ACB stands on chord AB in the specified alternate segment.
2
Apply the tangent–chord theorem.
$∠ A C B = 52^{\circ}$
The stated tangent–chord angle equals the circumference angle in that alternate segment.

Answer

52°

03 / YOUR TURN

Try it yourself.

Check your understanding

A cyclic quadrilateral has an interior angle 73°. Find its opposite angle.

Show answer & reasoning

107°

In a cyclic quadrilateral, opposite interior angles total $180^{\circ}$ . The required angle is $180^{\circ} - 73^{\circ} = 107^{\circ}$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

ABCD is cyclic and $∠ A = 112^{\circ}$ . Find $∠ C$ .
2 marks
A simple quadrilateral has opposite angles $96^{\circ}$ and $84^{\circ}$ . What can be concluded?
2 marks
A tangent–chord angle is $41^{\circ}$ and C lies in its alternate segment. Find the corresponding circumference angle.
2 marks
A cyclic quadrilateral has opposite angles $3 x + 10$ and $5 x - 6$ degrees. Find $x$ and both angles.
3 marks
ABCD is cyclic. An exterior angle at A is $65^{\circ}$ . Find interior $∠ C$ and explain.
3 marks
A radius meets a tangent at A. Chord AB makes $35^{\circ}$ with radius AO. Find the smaller angle between the tangent line and chord line.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

ABCD is cyclic and $∠ A = 112^{\circ}$ . Find $∠ C$ .
2 marks
$180 - 112 = 68$ .
$68^{\circ}$
A simple quadrilateral has opposite angles $96^{\circ}$ and $84^{\circ}$ . What can be concluded?
2 marks
The opposite angles sum to $180^{\circ}$ .
It is cyclic.
A tangent–chord angle is $41^{\circ}$ and C lies in its alternate segment. Find the corresponding circumference angle.
2 marks
Use the alternate-segment theorem.
$41^{\circ}$
A cyclic quadrilateral has opposite angles $3 x + 10$ and $5 x - 6$ degrees. Find $x$ and both angles.
3 marks
$8 x + 4 = 180$ gives $x = 22$ .
$3 \times 22 + 10 = 76$ ; $5 \times 22 - 6 = 104$ .
$x = 22$ ; $76^{\circ}$ and $104^{\circ}$ .
ABCD is cyclic. An exterior angle at A is $65^{\circ}$ . Find interior $∠ C$ and explain.
3 marks
Interior $A = 115^{\circ}$ .
Opposite $C = 180 - 115 = 65$ .
$65^{\circ}$
A radius meets a tangent at A. Chord AB makes $35^{\circ}$ with radius AO. Find the smaller angle between the tangent line and chord line.
2 marks
Radius and tangent make $90^{\circ}$ .
The remaining angle is $90 - 35 = 55$ .
$55^{\circ}$

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.

Official curriculum source ↗

In this lesson

Understand the idea.

Four vertices on one circle

Tangent and chord

See the method in action.

Cyclic quadrilateral

Pair the opposite vertices.

Calculate angle C.

Calculate angle D.

Tangent and chord

Match chord AB to the alternate-segment angle.

Apply the tangent–chord theorem.

Try it yourself.

Check your understanding

Keep practising